287 research outputs found
Approximating spectral invariants of Harper operators on graphs II
No full textWe study Harper operators and the closely related discrete magnetic Laplacians (DML) on a graph with a free action of a discrete group, as defined by Sunada. The spectral density function of the DML is defined using the von Neumann trace associated with the free action of a discrete group on a graph. The main result in this paper states that when the group is amenable, the spectral density function is equal to the integrated density of states of the DML that is defined using either Dirichlet or Neumann boundary conditions. This establishes the main conjecture in a paper by Mathai and Yates. The result is generalized to other self adjoint operators with finite propagation speed
Approximating L2 invariants and the Atiyah conjecture
No full textThe definitive version may be found at www.wiley.comAbstractLet G be a torsion‐free discrete group, and let ℚ denote the field of algebraic numbers in ℂ. We prove that ℚG fulfills the Atiyah conjecture if G lies in a certain class of groups D, which contains in particular all groups that are residually torsion‐free elementary amenable or are residually free. This result implies that there are no nontrivial zero divisors in ℂG. The statement relies on new approximation results for L2‐Betti numbers over ℚG, which are the core of the work done in this paper. Another set of results in the paper is concerned with certain number‐theoretic properties of eigenvalues for the combinatorial Laplacian on L2‐cochains on any normal covering space of a finite CW complex. We establish the absence of eigenvalues that are transcendental numbers whenever the covering transformation group is either amenable or in the Linnell class . We also establish the absence of eigenvalues that are Liouville transcendental numbers whenever the covering transformation group is either residually finite or more generally in a certain large bootstrap class . © 2003 Wiley Periodicals, Inc.Józef Dodziuk, Peter Linnell, Varghese Mathai, Thomas Schick, Stuart Yate
Von Neumann spectra near the spectral gap
No full textCarey, Alan L. ; Coulhon, Thierry ; Mathai, Varghese ; Phillips, Joh
T-duality of current algebras and their quantization
No full textIn this paper we show that the T-duality transform of Bouwknegt, Evslin and Mathai applies to determine isomorphisms of certain current algebras and their associated vertex algebras on topologically distinct T-dual spacetimes compactified to circle bundles with -flux.Pedram Hekmati, Varghese Matha
D-branes, RR-fields and duality on noncommutative manifolds
No full textWe develop some of the ingredients needed for string theory on noncommutative spacetimes, proposing an axiomatic formulation of T-duality as well as establishing a very general formula for D-brane charges. This formula is closely related to a noncom4 mutative Grothendieck-Riemann-Roch theorem that is proved here. Our approach relies on a very general form of Poincaré duality, which is studied here in detail. Among the technical tools employed are calculations with iterated products in bivariant K-theory and cyclic theory, which are simplified using a novel diagram calculus reminiscent of Feynman diagrams
Homotopy invariance of Novikov-Shubin invariants and L2 Betti numbers
No full textWe give short proofs of the Gromov-Shubin theorem on the homotopy invariance of the Novikov-Shubin invariants and of the Dodziuk theorem on the homotopy invariance of the Betti numbers of the universal covering of a closed manifold in this paper. We show that the homotopy invariance of these invariants is no more difficult to prove than the homotopy invariance of ordinary homology theory.Jonathan Block, Varghese Mathai and Shmuel Weinberger
T-duality simplifies bulk-boundary correspondence: the parametrised case
No full textWe state a general conjecture that T-duality simplifies a model for the bulk-boundary correspondence in the parametrised context. We give evidence that it is valid by proving it in a special interesting case, which is relevant both to String Theory and to the study of topological insulators with defects in Condensed Matter Physics.Keith C. Hannabuss, Varghese Mathai, and Guo Chuan Thian
Exotic twisted equivariant cohomology of loop spaces, twisted Bismut–Chern character and T-duality
No full textWe define exotic twisted T-equivariant cohomology for the loop space LZ of a smooth manifold Z via the invariant differential forms on LZ with coefficients in the (typically non-flat) holonomy line bundle of a gerbe, with differential an equivariantly flat superconnection. We introduce the twisted Bismut–Chern character form, a loop space refinement of the twisted Chern character form in Bouwknegt et al. (Commun Math Phys 228:17–49, 2002) and Mathai and Stevenson (Commun Math Phys 236:161–186, 2003), which represents classes in the completed periodic exotic twisted T-equivariant cohomology of LZ.We establish a localisation theorem for the completed periodic exotic twisted T-equivariant cohomology for loop spaces and apply it to establish T-duality in a background flux in type II String Theory from a loop space perspective.Fei Han, Varghese Matha
Arithmetic properties of eigenvalues of generalized Harper operators on graphs
No full textThe original publication is available at www.springerlink.comLet denote the field of algebraic numbers in A discrete group G is said to have the σ-multiplier algebraic eigenvalue property, if for every matrix A ∈ Md((G, σ)), regarded as an operator on l2(G)d, the eigenvalues of A are algebraic numbers, where σ ∈ Z2(G, ) is an algebraic multiplier, and denotes the unitary elements of . Such operators include the Harper operator and the discrete magnetic Laplacian that occur in solid state physics. We prove that any finitely generated amenable, free or surface group has this property for any algebraic multiplier σ. In the special case when σ is rational (σn=1 for some positive integer n) this property holds for a larger class of groups containing free groups and amenable groups, and closed under taking directed unions and extensions with amenable quotients. Included in the paper are proofs of other spectral properties of such operators.Józef Dodziuk, Varghese Mathai and Stuart Yate
Bundle gerbes and moduli spaces
No full textIn this paper, we construct the index bundle gerbe of a family of self-adjoint Dirac-type operators, refining a construction of Segal. In a special case, we construct a geometric bundle gerbe called the caloron bundle gerbe, which comes with a natural connection and curving, and show that it is isomorphic to the analytically constructed index bundle gerbe. We apply these constructions to certain moduli spaces associated to compact Riemann surfaces, constructing on these moduli spaces, natural bundle gerbes with connection and curving, whose 3-curvature represent Dixmier-Douady classes that are generators of the third de Rham cohomology groups of these moduli spaces. © 2011 Elsevier B.V.Peter Bouwknegt, Varghese Mathai and Siye Wuhttp://www.journals.elsevier.com/journal-of-geometry-and-physics
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